President Donald Trump’s personal attorney Michael Cohen helped a law and lobbying firm land a corporate client with ties to Kushner Cos., the family company of White House adviser Jared Kushner that is currently the subject of a federal probe.

Under a 2017 contract with Squire Patton Boggs, Mr. Cohen was paid $500,000 a year to help it drum up business, prosecutors said in a recent filing in federal court in New York, where Mr. Cohen is under investigation for potential bank fraud and campaign-finance violations. He also received a cut of any fees it collected from clients he referred.

Among five clients Mr. Cohen delivered to Squire Patton Boggs — before the firm terminated the contract with him in early March — was U.S. Immigration Fund LLC, according to court filings and Nicholas Mastroianni II, U.S. Immigration Fund’s chief executive. The Florida company connects businesses with foreign investors through a U.S. visa program.

After Mr. Trump’s election, Mr. Cohen began generating business both through his relationship with Squire Patton Boggs and through his own firm, Essential Consultants LLC.

Mr. Cohen formed Essential Consultants in October 2016 and used it to pay $130,000 to Stephanie Clifford, an adult-film actress known professionally as Stormy Daniels, to prevent her from publicly discussing an alleged sexual relationship with the president. Mr. Trump denies that relationship.

Federal agents and prosecutors in New York are examining whether Mr. Cohen violated any laws in his efforts to raise cash and conceal negative information about Mr. Trump, including the payment to Ms. Clifford, The Wall Street Journal has reported, citing people familiar with the matter. Federal Bureau of Investigation agents raided Mr. Cohen’s premises last month.

Special Counsel Robert Mueller, who is investigating Russian interference in the 2016 presidential elections, referred the Cohen investigation to federal prosecutors in Manhattan, the Journal previously reported.

Mr. Cohen didn’t respond to a request for comment.

U.S. Immigration Fund last year organized a trip to China for several Kushner Cos. officials, including Mr. Kushner’s sister Nicole Meyer, to seek investors for commercial-and-residential towers in Jersey City, N.J., the Journal has reported. Ms. Meyer pitched potential investors on participating in a program known as EB-5, which provides permanent U.S. residency to immigrants who invest at least $500,000 in certain job-creating businesses.

Kushner Cos. relied on U.S. Immigration Fund, which has pooled foreign funds for some of the largest developments in the EB-5 program, and a Chinese company to arrange the investor presentation. The project had been listed on U.S. Immigration Fund’s website, but the company ended its involvement with the Kushner Cos. development after the negative publicity from the presentation, Mr. Mastroianni said.

After company officials returned to the U.S., the Brooklyn U.S. attorney’s office and the U.S. Securities and Exchange Commission opened investigations into the Kushner Cos. use of EB-5, the Journal has reported.

Prosecutors in the Brooklyn U.S. attorney’s office have obtained video footage of Ms. Meyer’s investor presentation in China, which also featured a photo of Mr. Trump and references to the president and Mr. Kushner, according to people familiar with the matter. Though aspects of Ms. Meyer’s presentation were made public through news reports at the time, there was no full accounting of her remarks, and investigators were able to review her presentation by obtaining a cellphone video from her husband, who also attended the trip.

Through its general counsel, Kushner Cos. has said it “fully complied with its rules and regulations and did nothing improper” with regard to the EB-5 program.

A spokesman for the Brooklyn U.S. attorney’s office, John Marzulli, said, “I can’t confirm or deny an investigation.”

Federal prosecutors handling the investigation suggested to Kushner Cos. in recent weeks that the company isn’t a target of their probe in the EB-5 case, according to a person briefed on the matter, and have told company officials that their involvement in the future will likely be limited to serving as possible witnesses.

People familiar with the Brooklyn U.S. attorney’s office, however, say the office doesn’t consider anyone a target of an investigation until the office is poised to charge them.

U.S. Immigration Fund had partnered with Kushner Cos. on another Jersey City project a few years earlier, raising $50 million for a luxury apartment complex that was spearheaded by Jared Kushner, according to U.S. Immigration Fund’s website. Mr. Kushner was CEO of the company until he resigned to accept his White House position.

Mr. Mastroianni, U.S. Immigration Fund’s CEO, said in an interview that he met with Mr. Cohen in the attorney’s Rockefeller Center office last year. Mr. Cohen put him in touch with Edward Newberry, a top lobbyist in the Washington, D.C., office of Squire Patton Boggs, Mr. Mastroianni said.

Squire Patton Boggs has received $370,000 from U.S. Immigration Fund to lobby Congress and the administration on reforming the EB-5 visa system, according to lobbying reports. The firm hasn’t represented the company on any legal matters, according to a firm spokesman.

U.S. Immigration Fund and developers have beat back efforts by lawmakers to change EB-5 rules to ban the program’s use to finance projects in wealthier areas instead of in rural and high-unemployment areas.

Squire Patton Boggs described its work for the company as lobbying on “reform of the immigrant investor EB-5 program,” in the firm’s June 2017 lobbying registration statement.

Mr. Mastroianni donated $150,000 last year to a joint fundraising committee in support of Mr. Trump’s re-election, according to Federal Election Commission records. Mr. Mastroianni’s family also gave $100,000 to Mr. Trump’s inauguration fund, records show.

Separately, it emerged Tuesday that Mr. Cohen’s own firm, Essential Consultants, received $500,000 in 2017 from an investment fund linked to a Russian oligarch and hundreds of thousands of dollars more from companies including AT&T Inc. and Novartis AG, according to a person familiar with the matter and spokespeople for the companies.

The payments were revealed in a memo released Tuesday by Michael Avenatti, a lawyer for Ms. Clifford. Mr. Avenatti declined to comment.

It is unclear where Mr. Avenatti got the information about Mr. Cohen’s bank transactions. The Treasury Department’s inspector general is investigating whether suspicious-activity reports related to Mr. Cohen were “improperly disseminated,” a spokesman for that office said Wednesday.

Under federal law, banks are required to flag transactions that appear to have no business or apparent lawful purpose or that deviate inexplicably from a customer’s normal bank activity. Suspicious-activity reports are filed to the Treasury’s Financial Crimes Enforcement Network, where federal investigators can access them.

It is a crime to disclose the existence of a suspicious-activity report or related information to any third parties.

AT&T Inc. said Tuesday it paid Essential Consultants in 2017 for “insights” into the Trump administration at a time when the telecommunications giant needed government approval for an $85 billion takeover of Time Warner Inc.

Mr. Avenatti’s memo also alleged that Novartis AG made several payments to Mr. Cohen. The pharmaceutical company said in a statement that it entered a one-year contract with Mr. Cohen’s Essential Consultants in February 2017, and paid it $1.2 million. He was to provide advice related to U.S. health-care policy matters, the company said, adding that it cooperated with the special counsel’s request for information last November.

Write to Erica Orden at erica.orden@wsj.com, Joe Palazzolo at joe.palazzolo@wsj.com and Julie Bykowicz at julie.bykowicz@wsj.com

Breaking the story

Erica Orden, Joe Palazzolo and Julie Bykowicz broke the news Michael Cohen, the personal attorney for President Donald Trump, helped a law and lobbying firm land a corporate client with ties to the family company of White House adviser Jared Kushner. According to a recent federal court filing, Mr. Cohen was paid $500,000 a year to bring businesses to Squire Patton Boggs and one such client was U.S. Immigration Fund LLC, which has ties to Jared Kushner Cos.

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